Re: Interval in a continuum is a fractal

On Nov 30, 6:13 pm, Venkat Reddy <vred...@xxxxxxxxx> wrote:
On Dec 1, 12:34 am, William Hughes <wpihug...@xxxxxxxxxxx> wrote:





On Nov 29, 10:07 pm, Venkat Reddy <vred...@xxxxxxxxx> wrote:

This is another perspective to explain things when I say you can't
break the conituum interval into points.

An interval in continuum such as time span is bounded by two time
instants, which are modeled as real numbers in mathematics. The
interval also has a non-zero extent or magnitude or span which is
modeled by the "measure" in mathematics.

Any statement that is even remotely related to "such an interval is
made up of certain things", has to deal with division process of such
interval. Simply defining that such interval is made of time instants
is like saying I just said, and you can't question. We need to attempt
a division process to see if it is right.

Another "ex cathedra" statement. Note: Even if you managed to define
"division
process" (you have given a hazy idea of what you mean, but many
details
are not specified. E.g. can a division process have infinitely many
steps),
we still don't know how to tell "if it is right". Something to do
with
not being able to divide into pieces of zero length, but since we
don't
even know that a "division process" can divide the line into its
components we can't even conclude that if a line is made of points
the points must have nonzero extent.

Still. Let's assume that a set of points cannot pass your "division
process" test (whatever that is). Consider instead the
psuedo-line consisting of the set of real numbers. We have
psuedo-intervals [a,b] and (a,b) which are subsets of the real
numbers.
We have a process called psuedo-cuts, by which we identify two
subsets, both of which are psuedo-intervals, We have a psuedo-extent
for which the subset [a,b] has extent b-a, and a real number
has extent 0. Since [a,b] has finite extent and is a set of points,
we immediately have that a non zero extent can be formed of parts
all of which have zero extent.

Note, there is nothing you can do with your "real" line, the
one that is not made of points, that cannot be done with the
psuedo-line, other than to say "the real line is not made of points".
This may give you a warm fuzzy, but I can't see any other use.
If you point out that the psuedo-line is only a mathematical object
the answer will be "Well DUH!".

That's a lot of progress on your part in separating natural continuum
from real line. Because it really looks bad when people insist that
reals ARE the continuum.


I think you seem to be talking about two separate meanings of
the word "continuum". One is the natural continuum, or physical
continuum, but the other is the set of reals (R). Both are
equally valid meanings of the word, continuum.

I agree that there is no practical task that we can't do using this
"approximate model" of natural continuum using reals. But is this what
pure mathematics is all about? Providing some approximate thing that
just works? Fine for now, but we need to be consciously aware of this
separation. That has been my point all the while.


What pure mathematics is about is considering abstracted
structures without dragging in the physical world. And in
that frame of reference, continuum *never* means the
physical continuum. It means R.

However, if we return back to the problem, why can't we improve our
model? Why were we forced to collapse the fractal pattern into a dust
of zero extent points? What could happen if we stayed at our boundary
until which we are confident that our logical truths still work?


What does that mean, stay at the boundary? OK, so it doesn't
collapse to zero extent points. So what's the extent?

What if we just conclude that real number line is made of intervals
separated by real numbers? This is the only thing we can confidently
say. Going to the extent of demolishing intervals and saying that they
are made of points have no basis or need. Whether there intervals
diminish to a point is unknown, if we consider reaching infinite is
impossible. So better not get into that zone of imagination which will
consume everyone's energy for ever without any result. We simply do
not have enough logical truths or imagination power to talk in that
realm.


The very *definition* of interval is a type of set of points.
It is a set of reals (which are *defined* as the "points" in R),
that are between two other reals, using the orders given on
R. In general, an interval can be defined on any ordered
set as those elements between two others, with respect to
the given order.

Why must R have elements other than real numbers in it?

So the model "intervals separated by real numbers" seems fine at any
scale. Between any two given different real numbers we have an
interval of non-zero extent. And real line is made of these non-zero
extents. Thats pretty much we can say.

The immediate impact is the set notion of real numbers is gone. An
ordered set requires adjacency. Saying that the members are ordered
but there are no adjacent members is ridiculous by any perspective and
contradictory to the general definition of order. It is like saying
the emperor has clothes on.


Actually, adjacency is *not* required by ordered sets. That
is *YOUR* axiom for orders, not that of Pure Mathematics.

See the "general definition of order":
http://en.wikipedia.org/wiki/Partial_order
http://en.wikipedia.org/wiki/Total_order

Note the properties given well. There is *no* axiom
that looks like this:

"Adjacency: Given any element a of the set, there exists
another element b not equal to a such that a <= b imples
that no element c exists such that a <= c and c <= b
that is not equal to a or b."

That is YOUR axiom, not that of Mathematics.

You might have twisted the definition for "order" using your math
lingo, but that's just yet another inefficient way of dealing with it
by fixing the symptoms than root cause.


Nah, YOU've twisted the definition by throwing in axioms
of "adjacency".

.

Relevant Pages

  • Re: Why real numbers and points cant model continuum
    ... Continuum, or are "a" continuum. ... not a model of it but the reals ... one must accept his intuitions, in a sense "work within his ... to find a flaw in classical mathematics he must first learn it! ...
    (sci.math)
  • Re: Is one-to-one mapping valid for comparing infinite-sized sets?
    ... continuum is being cut into parts. ... Since the fractal nature of "the continuum" as the set of reals is ... Only in the sense that all mathematics is "unnatural" by being ... a plane instead of line? ...
    (sci.math)
  • Re: Why real numbers and points cant model continuum
    ... Continuum, or are "a" continuum. ... not a model of it but the reals ... one must accept his intuitions, in a sense "work within his ... to find a flaw in classical mathematics he must first learn it! ...
    (sci.math)
  • Re: Skolems Paradox and why is math the way it is?
    ... This is not a job the axioms were ever meant to do. ... It's actually a problem with language generally, not just mathematics. ... just procrastinated the problem of interpretation for one step. ... Consider for example the set of reals you considered in another posting, ...
    (sci.math)
  • Re: Why real numbers and points cant model continuum
    ... Attempt to "model" the continuum using real numbers? ... not a model of it but the reals ... one must accept his intuitions, in a sense "work within his ... to find a flaw in classical mathematics he must first learn it! ...
    (sci.math)